where \(\mathbf {h}_{\mathrm {d},k}\in \mathbb {C}^{N}\) , \(\mathbf {h}_{\mathrm {r},k} \in \mathbb {C}^{M}\)  denote the channels from the BS to the \(k\) -th user, from the RIS to the \(k\) -th user, respectively
With various advanced algorithms introduced in [1]}-[2]}
and the references therein to obtain the CSI for RIS-assisted systems, we assume perfect CSI in this paper in order to focus on exploring the potential of the reflection and transmission dual-functional active RIS-assisted system.,
and \(n_{k} \sim \mathcal {CN}\left(0, \sigma _k^{2}\right)\)  denotes the complex additive white Gaussian noise with variance \(\sigma _k^{2}\)  at the \(k\) -th user. Thus, the signal-to-interference-plus-noise ratio (SINR) of the \(k\) -th user, \(k\in \mathcal {K}_\mathrm {r}\) , can be expressed as
\(\begin{aligned}\mathrm {SINR}_{k}=& \frac{\left|\widetilde{\mathbf {h}}_{k}^{H}\mathbf {w}_{k}\right|^2}{\sum _{j \ne k}^{K}\left|\widetilde{\mathbf {h}}_{k}^{H}\mathbf {w}_{j} \right|^{2}+\sigma _{v}^{2}\Vert \mathbf {h}_{\mathrm {r},k}^{H}\mathbf {\Phi }_{\mathrm {r}}\mathbf {E}_{\mathrm {r}}\mathbf {A}\Vert ^2+{\sigma _k^{2}}}\\\stackrel{(a)}{=}&\frac{\left|\widetilde{\mathbf {h}}_{k}^{H}\mathbf {w}_{k}\right|^2}{\sum _{j \ne k}^{K}\left|\widetilde{\mathbf {h}}_{k}^{H}\mathbf {w}_{j} \right|^{2}+\sigma _{v}^{2}\Vert \mathbf {h}_{\mathrm {r},k}^{H}\mathbf {E}_{\mathrm {r}}\mathbf {A}\Vert ^2+{\sigma _k^{2}}},~k \in \mathcal {K}_\mathrm {r},\end{aligned}\) 
